| Many evolutionary partial differential equations may be rewritten as the gradient flow of an energy functional, a perspective which provides useful estimates on the behavior of solutions. The notion of gradient flow requires both the specification of an energy functional and a metric with respect to which the gradient is taken. In recent years, there has been significant interest in gradient flow on the space of probability measures endowed with the Wasserstein metric. The notion of gradient in this setting in purely formal and rigorous analysis of the gradient flow typically considers a time discretization of the problem known as the discrete gradient flow. In this dissertation, we adapt Crandall and Liggett's Banach space method to give a new proof of the exponential formula, quantifying the rate at which solutions to the discrete gradient flow converge to solutions of the gradient flow. In the process, we use a new class of metrics--- transport metrics---that have stronger convexity properties than the Wasserstein metric to prove an Euler-Lagrange equation characterizing the discrete gradient flow. We then apply these results to give simple proofs of properties of the gradient flow, including the contracting semigroup property and the energy dissipation inequality. |
